Question

Qual seria a 1ª derivada da função f(x)=sen[e^√(x^7+x)] ? Obrigado desde já ;P

Answer 1

$\mathrm{y=\sin({e^{\sqrt{x^7+x}}})}\\\\ \textrm{A partir da regra da cadeia, teremos que:}\\\\ \mathrm{\dfrac{dy}{dx}=\dfrac{d}{dx}(\sin({e^{\sqrt{x^7+x}}})).\dfrac{d}{dx}(e^{\sqrt{x^7+x}})}\\\\\\ \mathrm{\dfrac{dy}{dx}=\cos({e^{\sqrt{x^7+x}}}).e^{\sqrt{x^7+x}}.\dfrac{d}{dx}(\sqrt{x^7+x})}\\\\\\ \mathrm{\dfrac{dy}{dx}=\cos({e^{\sqrt{x^7+x}}}).e^{\sqrt{x^7+x}}.\dfrac{1}{2}.(x^7+x)^{-\frac{1}{2}}.\dfrac{d}{dx}(x^7+x)}$

$\mathrm{\dfrac{dy}{dx}=\cos({e^{\sqrt{x^7+x}})}.e^{\sqrt{x^7+x}}.\dfrac{1}{2\sqrt{x^7+x}}.(7x^6+1)}\\\\\\ \mathbf{\dfrac{dy}{dx}=\dfrac{(7x^6+1).e^{\sqrt{x^7+x}}.\cos({e^{\sqrt{x^7+x}}})}{2\sqrt{x^7+x}}}$